Following a solid opening set, the third seed from Romania served for the match in the second but lost the plot and her Polish opponent seized the opportunity to push the tie into a decider.
Halep, however, regained her composure to set up a meeting with Ukraine's Lesia Tsurenko or Serbian Aleksandra Krunic.
"I was close in the end of the second set, but she played really well, she was unbelievable, I'm happy I could win this one," said Halep.
"I tried everything but I didn't feel that great, I was a bit sick. I gave everything I had on court.
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"Tomorrow I will sleep all day because I am a bit sick."
Linette had brought with her some strategy notes on a piece of paper and she kept Halep in check throughout with some aggressive tennis.
Halep broke for 3-2 in the first set and she held serve to take the lead, but her game was still riddled with unforced errors.
She went 5-4 up in the second, only for Linette to break back and then break again in the 12th to force a third set.
Her first serve, however, then went AWOL, allowing Halep to secure the win.
(Reporting by Julien Pretot; Editing by Toby Davis)
